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{{Short description|Type of sequence of prime numbers}}
{{pp-semi-indef|small=yes}}
In [[mathematics]], a '''Cunningham chain''' is a certain [[integer sequence|sequence]] of [[prime number]]s. Cunningham chains are named after [[mathematician]] [[Allan Joseph Champneys Cunningham|A. J. C. Cunningham]]. They are also called '''chains of nearly doubled primes'''.

==Definition==

A '''Cunningham chain of the first kind''' of length ''n'' is a sequence of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that ''p''<sub>''i''+1</sub>&nbsp;=&nbsp;2''p''<sub>''i''</sub>&nbsp;+&nbsp;1 for all 1&nbsp;≤&nbsp;''i''&nbsp;<&nbsp;''n''. (Hence each term of such a chain except the last is a [[Sophie Germain prime]], and each term except the first is a [[safe prime]]).

It follows that

: <math>
\begin{align}
p_2 & = 2p_1+1, \\
p_3 & = 4p_1+3, \\
p_4 & = 8p_1+7, \\
& {}\  \vdots \\
p_i & = 2^{i-1}p_1 + (2^{i-1}-1),
\end{align}
</math>

or, by setting <math>a = \frac{p_1 + 1}{2}</math> (the number <math>a</math> is not part of the sequence and need not be a prime number), we have <math>p_i = 2^{i} a - 1.</math>

Similarly, a '''Cunningham chain of the second kind''' of length ''n'' is a sequence of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that ''p''<sub>''i''+1</sub>&nbsp;=&nbsp;2''p''<sub>''i''</sub>&nbsp;−&nbsp;1 for all 1&nbsp;≤&nbsp;''i''&nbsp;<&nbsp;''n''.

It follows that the general term is

: <math>p_i = 2^{i-1}p_1 - (2^{i-1}-1).</math>

Now, by setting <math>a = \frac{p_1 - 1}{2} </math>, we have <math> p_i = 2^{i} a + 1</math>.

Cunningham chains are also sometimes generalized to sequences of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that ''p''<sub>''i''+1</sub> =&nbsp;''ap''<sub>''i''</sub>&nbsp;+&nbsp;''b'' for all 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n'' for fixed [[coprime]] [[integer]]s ''a'' and ''b''; the resulting chains are called '''generalized Cunningham chains'''.

A Cunningham chain is called '''complete''' if it cannot be further extended, i.e., if the previous and the next terms in the chain are not prime numbers.

==Examples==

Examples of complete Cunningham chains of the first kind include these:

: 2, 5,&thinsp;11, 23, 47 (The next number would be 95, but that is not prime.)
: 3, 7 (The next number would be 15, but that is not prime.)
: 29, 59 (The next number would be 119, but that is not prime.)
: 41, 83, 167 (The next number would be 335, but that is not prime.)
: 89, 179, 359, 719, 1439, 2879 (The next number would be 5759, but that is not prime.)

Examples of complete Cunningham chains of the second kind include these:

: 2, 3, 5 (The next number would be 9, but that is not prime.)
: 7, 13 (The next number would be 25, but that is not prime.)
: 19, 37, 73 (The next number would be 145, but that is not prime.)
: 31, 61 (The next number would be 121 = 11<sup>2</sup>, but that is not prime.)

Cunningham chains are now considered useful in cryptographic systems since "they provide two concurrent suitable settings for the [[ElGamal encryption|ElGamal cryptosystem]] ... [which] can be implemented in any field where the [[discrete logarithm problem]] is difficult."<ref>Joe Buhler, ''Algorithmic Number Theory: Third International Symposium, ANTS-III''. New York: Springer (1998): 290</ref>

== Largest known Cunningham chains ==

It follows from [[Dickson's conjecture]] and the broader [[Schinzel's hypothesis H]], both widely believed to be true, that for every ''k'' there are infinitely many Cunningham chains of length ''k''.  There are, however, no known direct methods of generating such chains.

There are computing competitions for the longest Cunningham chain or for the one built up of the largest primes, but unlike the breakthrough of [[Ben J. Green]] and [[Terence Tao]] – the [[Green–Tao theorem]], that there are arithmetic progressions of primes of arbitrary length – there is no general result known on large Cunningham chains to date.

{| class="wikitable sortable"
|+ Largest known Cunningham chain of length ''k'' (as of 18 February 2025<ref name="records">Norman Luhn & Dirk Augustin, [https://www.pzktupel.de/JensKruseAndersen/cc.htm''Cunningham Chain records'']. Retrieved on 2025-02-18.</ref>)
|-
! ''k'' !! Kind !! ''p''<sub>1</sub> (starting prime) !! Digits !! Year !! Discoverer
|-
| 1 || 1st / 2nd || 2<sup>136279841</sup> − 1 || align="right" | 41024320 || 2024 || Luke Durant, [[Great Internet Mersenne Prime Search|GIMPS]]
|-
| rowspan="2" | 2 || 1st || 2618163402417×2<sup>1290000</sup> − 1 || align="right" | 388342 || 2016 || [[PrimeGrid]]
|-
| 2nd || 213778324725×2<sup>561417</sup> + 1 || align="right" | 169015 || 2023 || Ryan Propper & Serge Batalov
|-
| rowspan="2" | 3 || 1st || 1128330746865×2<sup>66439</sup> − 1 || align="right" | 20013 || 2020 || Michael Paridon
|-
| 2nd || 214923707595×2<sup>49073</sup> + 1 || align="right" | 14784 || 2025 || Serge Batalov
|-
| rowspan="2" | 4 || 1st || 93003628384×10111# − 1 || align="right" | 4362 || 2025 || Serge Batalov
|-
| 2nd || 49325406476×9811# + 1	 || align="right" | 4234|| 2019 || Oscar Östlin
|-
| rowspan="2" | 5 || 1st || 475676794046977267×4679# − 1 || align="right" | 2019 || 2024 || Andrey Balyakin
|-
| 2nd || 181439827616655015936×4673# + 1 || align="right" | 2018 || 2016 || Andrey Balyakin
|-
| rowspan="2" | 6 || 1st || 2799873605326×2371# − 1 || align="right" | 1016 || 2015 || Serge Batalov
|-
| 2nd || 37015322207094×2339# + 1 || align="right" | 1001 || 2025 || Serge Batalov
|-
| rowspan="2" | 7 || 1st || 82466536397303904×1171# − 1 || align="right" | 509 || 2016 || Andrey Balyakin
|-
| 2nd || 25802590081726373888×1033# + 1 || align="right" | 453 || 2015 || Andrey Balyakin
|-
| rowspan="2" | 8 || 1st || 89628063633698570895360×593# − 1 || align="right" | 265 || 2015 || Andrey Balyakin
|-
| 2nd || 2373007846680317952×761# + 1 || align="right" | 337 || 2016 || Andrey Balyakin
|-
| rowspan="2" | 9 || 1st || 553374939996823808×593# − 1 || align="right" | 260 || 2016 || Andrey Balyakin
|-
| 2nd || 173129832252242394185728×401# + 1 || align="right" | 187 || 2015 || Andrey Balyakin
|-
| rowspan="2" | 10 || 1st || 3696772637099483023015936×311# − 1 || align="right" | 150 || 2016 || Andrey Balyakin
|-
| 2nd || 2044300700000658875613184×311# + 1 || align="right" | 150 || 2016 || Andrey Balyakin
|-
| rowspan="2" | 11 || 1st || 73853903764168979088206401473739410396455001112581722569026969860983656346568919×151# − 1 || align="right" | 140 || 2013 || Primecoin ([https://primes.zone/#records block 95569])
|-
| 2nd || 341841671431409652891648×311# + 1 || align="right" | 149 || 2016 || Andrey Balyakin
|-
| rowspan="2" | 12 || 1st || 288320466650346626888267818984974462085357412586437032687304004479168536445314040×83# − 1 || align="right" | 113 || 2014 || Primecoin ([https://primes.zone/#records block 558800])
|-
| 2nd || 906644189971753846618980352×233# + 1 || align="right" | 121 || 2015 || Andrey Balyakin
|-
| rowspan="2" | 13 || 1st || 106680560818292299253267832484567360951928953599522278361651385665522443588804123392×61# − 1 || align="right" | 107 || 2014 || Primecoin ([https://primes.zone/#records block 368051])
|-
| 2nd || 38249410745534076442242419351233801191635692835712219264661912943040353398995076864×47# + 1 || align="right" | 101 || 2014 || Primecoin ([https://primes.zone/#records block 539977])
|-
| rowspan="2" | 14 || 1st || 4631673892190914134588763508558377441004250662630975370524984655678678526944768×47# − 1  || align="right" | 97 || 2018 || Primecoin ([https://primes.zone/#records block 2659167])
|-
| 2nd || 5819411283298069803200936040662511327268486153212216998535044251830806354124236416×47# + 1 || align="right" | 100 || 2014 || Primecoin ([https://primes.zone/#records block 547276])
|-
| rowspan="2" | 15 || 1st || 14354792166345299956567113728×43# - 1 || align="right" | 45 || 2016 || Andrey Balyakin
|-
| 2nd || 67040002730422542592×53# + 1 || align="right" | 40 || 2016 || Andrey Balyakin
|-
| rowspan="2" | 16 || 1st || 91304653283578934559359 || align="right" | 23 || 2008 || Jaroslaw Wroblewski
|-
| 2nd || 2×1540797425367761006138858881 − 1 || align="right" | 28 || 2014 || Chermoni & Wroblewski
|-
| rowspan="2" | 17 || 1st || 2759832934171386593519 || align="right" | 22 || 2008 || Jaroslaw Wroblewski
|-
| 2nd || 1540797425367761006138858881 || align="right" | 28 || 2014 || Chermoni & Wroblewski
|-
| 18 || 2nd || 658189097608811942204322721 || align="right" | 27 || 2014 || Chermoni & Wroblewski
|-
| 19 || 2nd || 79910197721667870187016101 || align="right" | 26 || 2014 || Chermoni & Wroblewski
|}

''q''# denotes the [[primorial]] 2&thinsp;×&thinsp;3&thinsp;×&thinsp;5&thinsp;×&thinsp;7&thinsp;×&thinsp;...&thinsp;×&thinsp;''q''.

{{As of|2018}}, the longest known Cunningham chain of either kind is of length 19, discovered by Jaroslaw Wroblewski in 2014.<ref name="records"/>

== Congruences of Cunningham chains ==

Let the [[parity (mathematics)|odd]] prime <math>p_1</math> be the first prime of a Cunningham chain of the first kind. The first prime is odd, thus <math>p_1 \equiv 1 \pmod{2}</math>. Since each successive prime in the chain is <math>p_{i+1} = 2p_i + 1</math> it follows that <math>p_i \equiv 2^i - 1 \pmod{2^i}</math>. Thus, <math>p_2 \equiv 3 \pmod{4}</math>, <math>p_3 \equiv 7 \pmod{8}</math>, and so forth.

The above property can be informally observed by considering the primes of a chain in [[base 2]]. (Note that, as with all bases, multiplying by the base "shifts" the digits to the left; e.g. in decimal we have 314&thinsp;×&thinsp;10 = 3140.) When we consider &nbsp;<math>p_{i+1} = 2p_i + 1</math> in base 2, we see that, by multiplying &nbsp;<math>p_i</math> by 2, the least significant digit of &nbsp;<math>p_i</math> becomes the secondmost least significant digit of &nbsp;<math>p_{i+1}</math>. Because <math>p_i</math> is odd—that is, the least significant digit is 1 in base 2–we know that the secondmost least significant digit of &nbsp;<math>p_{i+1}</math> is also 1. And, finally, we can see that &nbsp;<math>p_{i+1}</math> will be odd due to the addition of 1 to <math>2p_i</math>. In this way, successive primes in a Cunningham chain are essentially shifted left in binary with ones filling in the least significant digits. For example, here is a complete length 6 chain which starts at 141361469:

{| border="1" align="center" class="wikitable"
! Binary || Decimal
|- align="right"
| 1000011011010000000100111101 || 141361469
|- align="right"
| 10000110110100000001001111011 || 282722939
|- align="right"
| 100001101101000000010011110111 || 565445879
|- align="right"
| 1000011011010000000100111101111 || 1130891759
|- align="right"
| 10000110110100000001001111011111 || 2261783519
|- align="right"
| 100001101101000000010011110111111 || 4523567039
|}

A similar result holds for Cunningham chains of the second kind.  From the observation that <math>p_1 \equiv 1 \pmod{2}</math> and the relation <math>p_{i+1} = 2 p_i - 1</math> it follows that <math>p_i \equiv 1 \pmod{2^i}</math>.  In binary notation, the primes in a Cunningham chain of the second kind end with a pattern "0...01", where, for each <math>i</math>, the number of zeros in the pattern for <math>p_{i+1}</math> is one more than the number of zeros for <math>p_i</math>.  As with Cunningham chains of the first kind, the bits left of the pattern shift left by one position with each successive prime.

Similarly, because <math> p_i = 2^{i-1}p_1 + (2^{i-1}-1) \, </math> it follows that <math>p_i \equiv 2^{i-1} - 1 \pmod{p_1}</math>. But, by [[Fermat's little theorem]], <math>2^{p_1-1} \equiv 1 \pmod{p_1}</math>, so <math>p_1</math> divides <math>p_{p_1}</math> (i.e. with <math> i = p_1 </math>). Thus, no Cunningham chain can be of infinite length.<ref>{{cite journal|last=Löh|first=Günter|title=Long chains of nearly doubled primes|journal=Mathematics of Computation|date=October 1989|volume=53|issue=188|pages=751–759|doi=10.1090/S0025-5718-1989-0979939-8|url=https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0979939-8/|doi-access=free}}</ref>

== See also ==
* [[Primecoin]], which uses Cunningham chains as a proof-of-work system
* [[Bi-twin chain]]
* [[Primes in arithmetic progression]]

==References==
<references/>

== External links ==
* [http://primes.utm.edu/glossary/page.php?sort=CunninghamChain The Prime Glossary: Cunningham chain]
* [https://primes.zone/ Primecoin discoveries (primes.zone): online database of primecoin findings with list of records and visualization]
* [https://web.archive.org/web/20030716201258/http://primes.utm.edu/links/theory/special_forms/Cunningham_chains/ PrimeLinks++: Cunningham chain]
* {{OEIS el|1=A005602|2=Smallest prime beginning a complete Cunningham chain of length n (of the first kind)}} -- the first term of the lowest complete Cunningham chains of the first kind of length&nbsp;''n'',&nbsp;for&nbsp;1&nbsp;≤&nbsp;''n''&nbsp;≤&nbsp;14
* {{OEIS el|1=A005603|2=Chains of length n of nearly doubled primes}} -- the first term of the lowest complete Cunningham chains of the second kind with length&nbsp;''n'',&nbsp;for&nbsp;1&nbsp;≤&nbsp;''n''&nbsp;≤&nbsp;15

{{Prime number classes}}

[[Category:Prime numbers]]